What the Numbers Say About Gender Differences: Data on abilities reveal a great deal of overlap for men and women

Writing below (from the WSJ), EUGENIA CHENG offers a good primer on the subject of sex differences in IQ.  When one gets little more than ill-informed raves from Leftists, Dr. Cheng is to be congratulated for her cool reason.

Her basic point is that differences in abilities do exist but are mostly too small to be important.  And she tackles the one question within the field that is undoubtedly important:  Differences in mathematical ability.  All psychometricians cheerfully agree that women are better in verbal ability.  Men regularly lose arguments with their wives. But the shortage of women in STEM fields does seem to be of concern, perhaps because of the high prestige of some jobs in those fields

So the big point in her article is the small difference in math ability scores shown in the findings by Janet Hyde. I am afraid, however that the Hyde findings cannot be accepted uncritically. Her findings make little allowance for the importance of age in these studies. Using the Progressive Matrices, Lynn and Irwing showed a vast gap in ability among adults but almost no difference among children up to age 14. Adolescents aged 15 to 19 were intermediate.

Most studies of mathematical ability do not separate out age in that way so much of the variability in their results can be attributed to that failure.  But the important thing is that  Hyde's reliance on a great grab-bag of ages in her tabulations makes her work largely irrelevant.  If she really wanted to get at adult differences, she should have studied adult differences alone.  Some genetic studies reveal in fact that genetic differences don't reveal themselves fully until around age 30.

At any event, to cut a long story short, if the age differences shown in the PM's are reflected in mathematical ability, we would expect only small differences between males and females in initial enrolments in mathematical courses, followed by very large achievement gaps in the long slow grind through university studies and into the professions.  And that is roughly what we find.  The male-female difference in mathematical ability is definitely non-trivial

But what makes it non-trivial?  In her final paragraph, Dr Cheng makes the challenging point that "it is logically flawed to infer a biological difference from a statistical difference". But if not that which?  The twin studies are certainly univocal:  The ability gap is inborn.  A genetic difference may not be inferable from the studies she quotes but many other studies do indicate a genetic basis for the differences

IN 2005 LAWRENCE SUMMERS, then the president of Harvard, caused an uproar by appearing to suggest that the lack of women professors in math and science might arise from biological differences. Fifteen years on, a gender imbalance in these fields persists and the arguments rage on. I believe math can help us to progress. The discipline of math involves, among other things, ironing out ambiguities and providing clear definitions for comparisons. Men and women are not homogeneous groups of people who all behave in the same way, so we need ways to understand whole sets of data.

Averages are one well-known way; we can compare how men and women do at something “on average.” There are different types of averages: The mean is where we add up all results and divide by the total number of people, and the median is the 50th percentile that tells us that half the people rank above it and half rank below. For example, the mean height of men in the U.S. is 5 feet 9 inches, and for women it is 5 inches less, but plenty of women are taller than plenty of men. Averages don’t tell us much about differences among entire sets of data because they neglect how widely the data are spread. That spread of data can be studied via the standard deviation, which is calculated from the distance that each data point ranges from the mean. For a standard bell curve, a distance of “one standard deviation” on ei- ther side of the mean always comprises a fixed proportion of the results, around 70%.

The standard deviation for height is around 2.5 inches, so the mean heights of men and women are about two standard deviations apart. Thus, around 95% of women are shorter than the average for men, but there is still a noticeable overlap. For data sets that differ by one standard deviation or less, there is more substantial overlap. Average marathon times for men and women differ by about 30 minutes, for instance. That sounds like a lot, but is only half the standard deviation of one hour—and the fastest women run marathons twice as fast as average men.

Height and running times are particularly easy to measure, but men and women have also been compared in broader areas of behavior, such as mathematical skills, aggression and self-esteem. In 2005, Jane Shibley Hyde collated a large collection of meta-analyses of these differences. In her book “Inferior,” Angela Saini sums up the results: “In every case, except for throwing distance and vertical jumping, females are less than one standard deviation apart from males. On many measures, they are less than a tenth of a standard deviation apart, which is indistinguishable in everyday life.” For example, “mathematics problem solving” was found to be better in men by just 0.08 standard deviations; interestingly, women were found to outperform men at “mathematics concepts” by 0.03 standard deviations.

Men showed more self-esteem by a range of 0.04 to 0.21 standard deviations, increasing through adolescence; they were found more likely to make “intrusive interruptions” by 0.33 standard deviations. The differences may be interesting, but they are also very small. The differences within each gender are greater than the differences between genders, so gender is not a good predictor of these behaviors. Such comparisons are blurred, of course, by issues beyond the reach of mathematics. Many of the behaviors studied are much harder to define and measure than height or marathon times and involve some mix of biological and sociological influences. But it is logically flawed to infer a biological difference from a statistical difference. Mathematics provides us with powerful tools, but we have to know their uses and limits.


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